Rayleigh-Ritz variation principle

Gross, E. K. U., Oliveira, L. N., Kohn, W., 1988a, Rayleigh-Ritz Variational Principle for Ensembles of Fractionally Occupied States , Phys. Rev. A, 37, 2805. 

The density functional theory for ensembles is based on the generalized Rayleigh-Ritz variational principle [7]. The eigenvalue problem of the Hamiltonian H is given by... 

Eq. (22) shows that the Coulomb fitting variation principle is a boimd fixim below, so improving the fitting basis raises the total energy. This behavior is the reverse of what h pens to the total energy as the orbital basis is augmented (because of the Rayleigh-Ritz variational principle). The difference can seem counter-intuitive to new users of the method. 

Both techniques are based on the Rayleigh-Ritz variational principle using energy-independent basis functions of the muffin-tin orbital type [1.21]. 

Instead of applying tail cancellation as in Sect.2.1 where we derived the KKR-ASA equations, one may use the linear combination of muffin-tin orbitals (5.27) directly in a variational procedure. This has the advantages that it leads to an eigenvalue problem and that it is possible to include non-muffin-tin perturbations to the potential. According to the Rayleigh-Ritz variational principle, one varies y to make the energy functional stationary, i.e. 

To do this, we use the Rayleigh-Ritz variational principle in connection with the radial trial function of arbitrary logarithmic derivative D at the sphere boundary defined by the linear combination... 

Applying the Rayleigh-Ritz variational principle, one obtains 4 H, ) < ( P //i )... 

For many years configuration interaction was regarded as the method of choice in describing electron correlation effeets in atoms and moleeules. The method is robust and systematic being firmly based on the Rayleigh-Ritz variational principle. The total electronie wavefimetion, is written as a linear eombination of A/ -electron determinantal functions, < >, ,... 

Komi, D. 1., T. Markovich, N. Maxwell, and E. R. Bittner. 2009. Supersymmetric quantum mechanics, excited state energies and wave functions, and the Rayleigh-Ritz variational principle a proof of principle study. Journal of Physical Chemistry A 113 (52) 15257. 

Many of the calculations of quantum chemistry are based on the Rayleigh-Ritz variation principle which states For any normalized, acceptable function 4>,... 

The proof of the Rayleigh-Ritz variation principle (Section 6-12) involves essentially two ideas. The first is that any function can be expanded into a linear combination of other functions that span the same function space. Thus, for example, exp(/ x) can be expressed as cos(fo) + i sin(fo). An exponential can also be written as a linear combination of powers of the argument ... 

This theorem follows readily from the 1 1 correspondence between E and p and the Rayleigh-Ritz variational principle. By incorporating the A-electron constraint on the... 

The variational approach received a major boost also when it was realised [79] that the simplest variational method - the Kohn variational principle, which is essentially the Rayleigh-Ritz variational principle for eigenvalues modified to incorporate scattering boundary conditions - is free of anomalous (i.e., spurious, unphysical) singularities if it is formulated with S-matrix type boundary conditions rather than standing wave boundary conditions as had been typically used previously. It is useful first to state the Kohn variational approach for the general inelastic scattering. Thus the variational expression for the S-matrix is... 

The KVP resembles the Rayleigh-Ritz variational principle [30] (RRVP). Indeed, if the scattering variational formulation were applied to a bound state problem, for which = 0 = Sji>r nr E) the KVP reduces to... 

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